Solve for $x$ : $ 2|x + 5| - 3 = 1|x + 5| + 3 $
Solution: Subtract $ {1|x + 5|} $ from both sides: $ \begin{eqnarray} 2|x + 5| - 3 &=& 1|x + 5| + 3 \\ \\ { - 1|x + 5|} && { - 1|x + 5|} \\ \\ 1|x + 5| - 3 &=& 3 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 1|x + 5| - 3 &=& 3 \\ \\ { + 3} &=& { + 3} \\ \\ 1|x + 5| &=& 6 \end{eqnarray} $ Simplify: $ |x + 5| = 6$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -6 $ or $ x + 5 = 6 $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -6 $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -6 \\ \\ {- 5} && {- 5} \\ \\ x &=& -6 - 5 \end{eqnarray} $ $ x = -11 $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = 6 $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& 6 \\ \\ {- 5} && {- 5} \\ \\ x &=& 6 - 5 \end{eqnarray} $ $ x = 1 $ Thus, the correct answer is $x = -11 $ or $x = 1 $.